In this talk, we shall consider the locally uniform weak convergence in time of a sequence of finite measures with time parameter. We then describe some properties of a topology inducing this convergence and equivalent conditions in terms of characteristic functions. Using them, we shall prove that a sequence of additive processes converges to some additive process in law as RCLL processes (i.e., as Skorokhod space valued random elements) if and only if their distributions (at time t) converge weakly locally uniformly in t. These results extend an established convergence theorem and are related to recent studies on infinitely divisible distributions in commutative and non-commutative probability as well. This talk is based on the ongoing work with Takahiro Hasebe (Hokkaido Univ.) and Ikkei Hotta (Yamaguchi Univ.)